Regression Analysis with Dependent Variable Censored

Situations frequently occur in practice where a regression analysis seems suitable but where the dependent variable, in some instances, is known only to be greater than a particular value which varies among individuals. Such a situation would arise, for example, if we were studying times to death for a group of patients with a specific disease and at the time of the analysis some patients were still alive. If we wish to do a regression analysis with time to death (or a function of time to death) as the dependent variable the survival time for each of the surviving patients would obviously be known to be greater than its value at the time of the analysis. Given this situation, the maximum likelihood solution for the regression coefficients is presented below. The solution is an extension of the method presented by Lea 1945, who adapted previous work of Bliss and Stevens 1937. Other related work has been published by Cohen 1950; 1955; 1957, Sampford 1952a,b; 1954, Gupta 1952, Halperin 1952, and Boag 1949, as well as others (see references in these papers). In these papers, when more than one variable was considered, multivariate normality was assumed. The present approach assumes that the X's are fixed. Lea's presentation and notation will be closely followed throughout this paper.